When I was 12, the school nurse yelled at me for not wearing a bra. I didn't feel any need for one at the time before she mentioned it. In hindsight, she didn't tell me about it because of my chest size- she was pissed off because it made the scoliosis testing more difficult for her. Rather than being sympathetic that I'd forgotten they were testing that day and had meant to wear my swimsuit under my shirt like I'd done every year beforehand, instead she decided to yell at me while I was half-naked. I basically left the room in tears and immediately got a bra just out of shame. I was never able to find comfortable bras, I always hated the way they made my clothes look, I don't know if I ever would have decided to wear them if left to my own devices- and don't see anything wrong with that- but it sure would have been nice for my decision to have been based on genuine interest rather than shame!
(If a school nurse really feels this is an issue that needs to be addressed: my opinion is that they should call the parents and leave it to them to handle. It is not the nurse's child, it is not the nurse's place to tell a child what they should or should not wear.)
If a doctor told me that my kid had to wear a bra, I'd address the medical concerns and try to find a solution that's most comfortable. Beyond that, I'll only raise concerns about modesty and otherwise leave the timing up to my children. My biggest concern is how to make it clear that they're free to get a bra whenever they're ready without coming off as pressuring them into one. If it seems like they may need one but aren't asking, I'm not sure how to make sure they aren't asking out of genuine disinterest rather than out of discomfort about raising the subject.